Problem Sheet: Week 13 - Quantum Mechanics

Chapter 1: The position of a quantum particle

Exercise 1.2

Consider a particle with the following wavefunction,

\psi(x) = \begin{cases} A\left(1+\frac{x}{2a}\right)& \text{if } -2a \leq x \leq 0, \\ A\left(1-\frac{x}{2a}\right)& \text{if } 0 \leq x \leq 2a, \\ 0 &\text{otherwise, } \end{cases}

where $A>0$ and $a>0$ are constants.

Normalise the wavefunction. That is, find the value of $A$ such that the wavefunction is normalised.
Sketch the wavefunction $\psi(x)$ , and the probability density $\pd(x)$ .
What is the probability that a measurement of the position of the particle will find it between $x = a$ and $x = 2a$ .

Exercise 1.3

Consider a particle with the following wavefunction,

\psi(x) = \begin{cases} \frac{1}{\sqrt{a}}e^{2\pi x i /a} & \text{if } 0 \leq x \leq a, \\ 0 &\text{otherwise. } \end{cases}

where $a > 0$ is a constant.

Sketch the wavefunction $\psi(x)$ , taking care to plot both the real and imaginary parts.
Show that this is a normalised wavefunction.
Sketch the probability density $\pd(x)$ .
Where will a measurement of the position of the particle find it on average? That is, what is the expected location of the particle?

Hint

Recall that for a probability density $\pd(x)$ , the expectation (or expected) value is defined by

\langle x \rangle = \int_{-\infty}^{\infty} x \pd(x) dx,

(1.23)

which is the average, or mean, position where the particle will be found.

Chapter 2: The momentum of a quantum particle

Exercise 2.5

Consider the particle from Exercise 1.1,

\psi(x) = \begin{cases} \frac{\sqrt{15}}{4\sqrt{a}}\left(1-\frac{x^2}{a^2}\right)& \text{if } |x| \leq a, \\ 0 &\text{otherwise. } \end{cases}

(1.22)

Show that the momentum wavefunction of the particle is

\tilde{\psi}(p) = \sqrt{\frac{15 a}{2\pi \hbar}}\left(\frac{\hbar^3}{a^3p^3}\sin (ap/\hbar) - \frac{\hbar^2}{a^2p^2}\cos(ap/\hbar)\right).

(2.32)

Hint

You will need to integrate by parts twice in order to obtain this result.

Sketch the wavefunction $\tilde{\psi}(p)$ and the probability density $\pd(p)$ .

What is the probability density for the particle to have momentum $p = \frac{\pi \hbar}{a}$ ?

Exercise 2.6

Consider two particles, the first of which has wavefunction $\psi(x)$ , and the second of which has wavefunction $\psi'(x)$ , related to $\psi(x)$ via

\psi'(x) = \psi(x)e^{ik_0x},

where $k_0$ is a real constant.

Write down the momentum wavefunction of the second particle (the momentum wavefunction associated to $\psi'(x)$ ).
Show that the momentum wavefunctions of the two particles are related via
$\tilde{\psi}'(p) = \tilde{\psi}(p-\hbar k_0).$

Hint

It may be useful to introduce the new variable $p' = p-\hbar k_0$ in your answer to part 1.

In a single plot, make representative sketches of $\pd(p) = |\tilde\psi(p)|^2$ and $\pd'(p) = |\tilde\psi'(p)|^2$ (e.g. assuming a generic ‘Bell-shaped’ momentum wavefunction).
Use your answers to part 2 and 3 to explain what property/properties of a particle change when we multiply the spatial wavefunction by $e^{ik_0x}$ .

Exercise 2.7

Consider a particle with the following momentum wavefunction,

\tilde{\psi}(p) = \begin{cases} \frac{1}{\sqrt{p_b - p_a}} & \text{if } p_a \leq p \leq p_b, \\ 0 &\text{otherwise. } \end{cases}

where $p_a < p_b$ . This wavefunction has the form of a ``box’', of width $\Delta = p_b - p_a$ and centre at $p_c = (p_a + p_b)/2$ .

Show that the spatial wavefunction $\psi(x)$ of the particle is

\psi(x) = \sqrt{\frac{\hbar}{2\pi (p_b - p_a)}} \frac{(e^{ip_bx/\hbar} - e^{ip_ax/\hbar})}{ix}.

By expressing $p_a$ and $p_b$ in terms of the centre $p_c$ and width $\Delta$ , show that the wavefunction can alternatively be written as

\psi(x) = \sqrt{\frac{\Delta}{2\pi\hbar}} e^{ip_c x/\hbar}\mathrm{sinc}(x\Delta/2\hbar),

where $\mathrm{sinc}(y) = \sin(y)/y$ .

(Tricky) Consider now a particle with a spatial wavefunction

\psi'(x) = \sqrt{\frac{\Delta}{2\pi\hbar}}\mathrm{sinc}(x\Delta /2\hbar).

Use Exercise 2.6 to write down (i.e. without calculating explicitly) the momentum wavefunction $\tilde{\psi}'(p)$ . What is the centre and width of this wavefunction?

Chapter 3: Incompatible observables & the Heisenberg uncertainty principle

Exercise 3.4

In this exercise we prove the canonical commutation relation. Consider an arbitrary wavefunction $\psi(x)$ .

Calculate $\op{X}\left(\op{P}\psi(x)\right)$ .
Calculate $\op{P}\left(\op{X}\psi(x)\right)$ .
Using your answers to parts (1) and (2), calculate $[\op{X},\op{P}]\psi(x)$ .
Explain why

[\hat{X},\hat{P}]\ket{\psi} = \infint \left([\op{X},\op{P}]\psi(x)\right)\ket{x} dx.

(3.12)

Finally, using parts (3) and (4), find $[\hat{X},\hat{P}]$ .

Exercise 3.5

Consider the particle from Exercise 1.1,

\psi(x) = \begin{cases} \frac{\sqrt{15}}{4\sqrt{a}}\left(1-\frac{x^2}{a^2}\right)& \text{if } |x| \leq a, \\ 0 &\text{otherwise. } \end{cases}

(1.22)

which we showed in Exercise 2.5 had momentum wavefunction

\tilde{\psi}(p) = \sqrt{\frac{15 a}{2\pi \hbar}}\left(\frac{\hbar^3}{a^3p^3}\sin (ap/\hbar) - \frac{\hbar^2}{a^2p^2}\cos(ap/\hbar)\right).

(2.32)

Show that both $\psi(x)$ and $\tilde{\psi}(p)$ are even functions.

Explain why this implies that both the average position and average momentum are zero,

\begin{align*} \langle x \rangle &= 0,& \langle p \rangle &= 0. \end{align*}

Calculate the average square position of the particle, $\langle x^2 \rangle$ , and the standard deviation, $\Delta x~=~\sqrt{\langle x^2\rangle - \langle x \rangle^2}$ .

It can be shown that the average square momentum of the particle is

\vph \langle p^2 \rangle = \frac{5\hbar^2}{2a^2}.

Calculate the standard deviation in the momentum of the particle, $\Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}$ .
Show that the particle satisfies the HUP. How close is the particle to saturating the bound of the uncertainty principle?